Planar Graphs Have Bounded Queue-Number∗
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:1908.08938V1 [Cs.DS] 23 Aug 2019 U ≺ W ≺ X ≺ V (Under the Same Assumptions As Above)
Mixed Linear Layouts: Complexity, Heuristics, and Experiments Philipp de Col, Fabian Klute, and Martin N¨ollenburg Algorithms and Complexity Group, TU Wien, Vienna, Austria [email protected],ffklute,[email protected] Abstract. A k-page linear graph layout of a graph G = (V; E) draws all vertices along a line ` and each edge in one of k disjoint halfplanes called pages, which are bounded by `. We consider two types of pages. In a stack page no two edges should cross and in a queue page no edge should be nested by another edge. A crossing (nesting) in a stack (queue) page is called a conflict. The algorithmic problem is twofold and requires to compute (i) a vertex ordering and (ii) a page assignment of the edges such that the resulting layout is either conflict-free or conflict-minimal. While linear layouts with only stack or only queue pages are well-studied, mixed s-stack q-queue layouts for s; q ≥ 1 have received less attention. We show NP-completeness results on the recognition problem of certain mixed linear layouts and present a new heuristic for minimizing conflicts. In a computational experiment for the case s; q = 1 we show that the new heuristic is an improvement over previous heuristics for linear layouts. 1 Introduction Linear graph layouts, in particular book embeddings [1,11] (also known as stack layouts) and queue layouts [8,9], form a classic research topic in graph drawing with many applications beyond graph visualization as surveyed by Dujmovi´cand Wood [5]. A k-page linear layout Γ = (≺; P) of a graph G = (V; E) consists of an order ≺ on the vertex set V and a partition of E into k subsets P = fP1;:::;Pkg called pages. -
Planar Graphs Have Bounded Queue-Number
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) Planar Graphs have Bounded Queue-Number Vida Dujmovic´∗, Gwenaël Joret†, Piotr Micek‡, Pat Morin§, Torsten Ueckerdt¶, David R. Wood ∗ School of Computer Science and Electrical Eng., University of Ottawa, Canada ([email protected]) † Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium ([email protected]) ‡ Faculty of Mathematics and Computer Science, Jagiellonian University, Poland ([email protected]) § School of Computer Science, Carleton University, Ottawa, Canada ([email protected]) ¶ Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany ([email protected]) School of Mathematics, Monash University, Melbourne, Australia ([email protected]) Abstract—We show that planar graphs have bounded queue- open whether every graph has stack-number bounded by a number, thus proving a conjecture of Heath, Leighton and function of its queue-number, or whether every graph has Rosenberg from 1992. The key to the proof is a new structural queue-number bounded by a function of its stack-number tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each [1,2]. part has a bounded number of vertices in each layer, and the Planar graphs are the simplest class of graphs where it is quotient graph has bounded treewidth. This result generalises unknown whether both stack and queue-number are bounded. for graphs of bounded Euler genus. Moreover, we prove that In particular, Buss and Shor [3] first proved that planar every graph in a minor-closed class has such a layered partition graphs have bounded stack-number; the best known upper if and only if the class excludes some apex graph. -
Planar Graph Theory We Say That a Graph Is Planar If It Can Be Drawn in the Plane Without Edges Crossing
Planar Graph Theory We say that a graph is planar if it can be drawn in the plane without edges crossing. We use the term plane graph to refer to a planar depiction of a planar graph. e.g. K4 is a planar graph Q1: The following is also planar. Find a plane graph version of the graph. A B F E D C A Method that sometimes works for drawing the plane graph for a planar graph: 1. Find the largest cycle in the graph. 2. The remaining edges must be drawn inside/outside the cycle so that they do not cross each other. Q2: Using the method above, find a plane graph version of the graph below. A B C D E F G H non e.g. K3,3: K5 Here are three (plane graph) depictions of the same planar graph: J N M J K J N I M K K I N M I O O L O L L A face of a plane graph is a region enclosed by the edges of the graph. There is also an unbounded face, which is the outside of the graph. Q3: For each of the plane graphs we have drawn, find: V = # of vertices of the graph E = # of edges of the graph F = # of faces of the graph Q4: Do you have a conjecture for an equation relating V, E and F for any plane graph G? Q5: Can you name the 5 Platonic Solids (i.e. regular polyhedra)? (This is a geometry question.) Q6: Find the # of vertices, # of edges and # of faces for each Platonic Solid. -
Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges
Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen Kobourov1, and Michael Schulz2 1 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆ 2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆ Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for when a pair of graphs whose union is homeomorphic to K5 or K3,3 can have an SEFE. This allows us to determine which (outer)planar graphs always an SEFE with any other (outer)planar graphs. In both cases, we provide efficient al- gorithms to compute the simultaneous drawings. Finally, we provide an linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has an SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale integration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can be used with multiple edge sets that each correspond to different edge colors or circuit layers. While the pairwise union of all edge sets may be nonplanar, a planar drawing of each layer may be possible, as crossings between edges of distinct edge sets are permitted. -
Four Results of Jon Kleinberg a Talk for St.Petersburg Mathematical Society
Four Results of Jon Kleinberg A Talk for St.Petersburg Mathematical Society Yury Lifshits Steklov Institute of Mathematics at St.Petersburg May 2007 1 / 43 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 / 43 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 2 / 43 4 Navigation in a Small World 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 2 / 43 5 Bursty Structure in Streams Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 2 / 43 Outline 1 Nevanlinna Prize for Jon Kleinberg History of Nevanlinna Prize Who is Jon Kleinberg 2 Hubs and Authorities 3 Nearest Neighbors: Faster Than Brute Force 4 Navigation in a Small World 5 Bursty Structure in Streams 2 / 43 Part I History of Nevanlinna Prize Career of Jon Kleinberg 3 / 43 Nevanlinna Prize The Rolf Nevanlinna Prize is awarded once every 4 years at the International Congress of Mathematicians, for outstanding contributions in Mathematical Aspects of Information Sciences including: 1 All mathematical aspects of computer science, including complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence. -
Arxiv:2107.04993V1 [Cs.DS] 11 Jul 2021
The Mixed Page Number of Graphs Jawaherul Md. Alam1 , Michael A. Bekos2 , Martin Gronemann3 , Michael Kaufmann2 , and Sergey Pupyrev4 1 Amazon Inc., Tempe, AZ, USA [email protected] 2 Institut f¨urInformatik, Universit¨atT¨ubingen,T¨ubingen,Germany fbekos,[email protected] 3 Theoretical Computer Science, Osnabr¨uck University, Osnabr¨uck, Germany [email protected] 4 Facebook, Inc., Menlo Park, CA, USA [email protected] Abstract. A linear layout of a graph typically consists of a total vertex order, and a partition of the edges into sets of either non-crossing edges, called stacks, or non-nested edges, called queues. The stack (queue) number of a graph is the minimum number of required stacks (queues) in a linear layout. Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. In this work we initiate the study of the mixed page number of a graph which corresponds to the minimum number of such sets. First, we study the edge density of graphs with bounded mixed page number. Then, we focus on complete and complete bipartite graphs, for which we derive lower and upper bounds on their mixed page number. Our findings indicate that combining stacks and queues is more powerful in various ways compared to the two traditional layouts. Keywords: Linear layouts · Mixed page number · Stacks and queues 1 Introduction Linear layouts of graphs form an important research topic, which has been studied in different areas and under different perspectives. As a matter of fact, several combinatorial optimization problems are defined by means of a measure over a linear layout of a graph (including the well- known cutwidth [1], bandwidth [9] and pathwidth [23]). -
Navigability of Small World Networks
Navigability of Small World Networks Pierre Fraigniaud CNRS and University Paris Sud http://www.lri.fr/~pierre Introduction Interaction Networks • Communication networks – Internet – Ad hoc and sensor networks • Societal networks – The Web – P2P networks (the unstructured ones) • Social network – Acquaintance – Mail exchanges • Biology (Interactome network), linguistics, etc. Dec. 19, 2006 HiPC'06 3 Common statistical properties • Low density • “Small world” properties: – Average distance between two nodes is small, typically O(log n) – The probability p that two distinct neighbors u1 and u2 of a same node v are neighbors is large. p = clustering coefficient • “Scale free” properties: – Heavy tailed probability distributions (e.g., of the degrees) Dec. 19, 2006 HiPC'06 4 Gaussian vs. Heavy tail Example : human sizes Example : salaries µ Dec. 19, 2006 HiPC'06 5 Power law loglog ppk prob{prob{ X=kX=k }} ≈≈ kk-α loglog kk Dec. 19, 2006 HiPC'06 6 Random graphs vs. Interaction networks • Random graphs: prob{e exists} ≈ log(n)/n – low clustering coefficient – Gaussian distribution of the degrees • Interaction networks – High clustering coefficient – Heavy tailed distribution of the degrees Dec. 19, 2006 HiPC'06 7 New problematic • Why these networks share these properties? • What model for – Performance analysis of these networks – Algorithm design for these networks • Impact of the measures? • This lecture addresses navigability Dec. 19, 2006 HiPC'06 8 Navigability Milgram Experiment • Source person s (e.g., in Wichita) • Target person t (e.g., in Cambridge) – Name, professional occupation, city of living, etc. • Letter transmitted via a chain of individuals related on a personal basis • Result: “six degrees of separation” Dec. -
Separator Theorems and Turán-Type Results for Planar Intersection Graphs
SEPARATOR THEOREMS AND TURAN-TYPE¶ RESULTS FOR PLANAR INTERSECTION GRAPHS JACOB FOX AND JANOS PACH Abstract. We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of m p 2 crossings has a partition into three parts C = S [ C1 [ C2 such that jSj = O( m); maxfjC1j; jC2jg · 3 jCj; and no element of C1 has a point in common with any element of C2. These results are used to obtain various properties of intersection patterns of geometric objects in the plane. In particular, we prove that if a graph G can be obtained as the intersection graph of n convex sets in the plane and it contains no complete bipartite graph Kt;t as a subgraph, then the number of edges of G cannot exceed ctn, for a suitable constant ct. 1. Introduction Given a collection C = fγ1; : : : ; γng of compact simply connected sets in the plane, their intersection graph G = G(C) is a graph on the vertex set C, where γi and γj (i 6= j) are connected by an edge if and only if γi \ γj 6= ;. For any graph H, a graph G is called H-free if it does not have a subgraph isomorphic to H. Pach and Sharir [13] started investigating the maximum number of edges an H-free intersection graph G(C) on n vertices can have. If H is not bipartite, then the assumption that G is an intersection graph of compact convex sets in the plane does not signi¯cantly e®ect the answer. -
A Decade of Lattice Cryptography
Full text available at: http://dx.doi.org/10.1561/0400000074 A Decade of Lattice Cryptography Chris Peikert Computer Science and Engineering University of Michigan, United States Boston — Delft Full text available at: http://dx.doi.org/10.1561/0400000074 Foundations and Trends R in Theoretical Computer Science Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 United States Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is C. Peikert. A Decade of Lattice Cryptography. Foundations and Trends R in Theoretical Computer Science, vol. 10, no. 4, pp. 283–424, 2014. R This Foundations and Trends issue was typeset in LATEX using a class file designed by Neal Parikh. Printed on acid-free paper. ISBN: 978-1-68083-113-9 c 2016 C. Peikert All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for in- ternal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The ‘services’ for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. -
Constraint Based Dimension Correlation and Distance
Preface The papers in this volume were presented at the Fourteenth Annual IEEE Conference on Computational Complexity held from May 4-6, 1999 in Atlanta, Georgia, in conjunction with the Federated Computing Research Conference. This conference was sponsored by the IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, in cooperation with the ACM SIGACT (The special interest group on Algorithms and Complexity Theory) and EATCS (The European Association for Theoretical Computer Science). The call for papers sought original research papers in all areas of computational complexity. A total of 70 papers were submitted for consideration of which 28 papers were accepted for the conference and for inclusion in these proceedings. Six of these papers were accepted to a joint STOC/Complexity session. For these papers the full conference paper appears in the STOC proceedings and a one-page summary appears in these proceedings. The program committee invited two distinguished researchers in computational complexity - Avi Wigderson and Jin-Yi Cai - to present invited talks. These proceedings contain survey articles based on their talks. The program committee thanks Pradyut Shah and Marcus Schaefer for their organizational and computer help, Steve Tate and the SIGACT Electronic Publishing Board for the use and help of the electronic submissions server, Peter Shor and Mike Saks for the electronic conference meeting software and Danielle Martin of the IEEE for editing this volume. The committee would also like to thank the following people for their help in reviewing the papers: E. Allender, V. Arvind, M. Ajtai, A. Ambainis, G. Barequet, S. Baumer, A. Berthiaume, S. -
Treewidth-Erickson.Pdf
Computational Topology (Jeff Erickson) Treewidth Or il y avait des graines terribles sur la planète du petit prince . c’étaient les graines de baobabs. Le sol de la planète en était infesté. Or un baobab, si l’on s’y prend trop tard, on ne peut jamais plus s’en débarrasser. Il encombre toute la planète. Il la perfore de ses racines. Et si la planète est trop petite, et si les baobabs sont trop nombreux, ils la font éclater. [Now there were some terrible seeds on the planet that was the home of the little prince; and these were the seeds of the baobab. The soil of that planet was infested with them. A baobab is something you will never, never be able to get rid of if you attend to it too late. It spreads over the entire planet. It bores clear through it with its roots. And if the planet is too small, and the baobabs are too many, they split it in pieces.] — Antoine de Saint-Exupéry (translated by Katherine Woods) Le Petit Prince [The Little Prince] (1943) 11 Treewidth In this lecture, I will introduce a graph parameter called treewidth, that generalizes the property of having small separators. Intuitively, a graph has small treewidth if it can be recursively decomposed into small subgraphs that have small overlap, or even more intuitively, if the graph resembles a ‘fat tree’. Many problems that are NP-hard for general graphs can be solved in polynomial time for graphs with small treewidth. Graphs embedded on surfaces of small genus do not necessarily have small treewidth, but they can be covered by a small number of subgraphs, each with small treewidth. -
Layouts of Graph Subdivisions
Layouts of Graph Subdivisions Vida Dujmovi´c1,2 and David R. Wood2,3 1 School of Computer Science, McGill University, Montr´eal, Canada [email protected] 2 School of Computer Science, Carleton University, Ottawa, Canada [email protected] 3 Department of Applied Mathematics, Charles University, Prague, Czech Republic Abstract. A k-stack layout (respectively, k-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex or- dering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The stack-number (respectively, queue-number, track-number) of a graph G, denoted by sn(G)(qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout. This paper studies stack, queue, and track layouts of graph sub- divisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n)to O(log min{sn(G), qn(G)}). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number. It is proved that every graph has a 2- queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision.